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arXiv · 2504.00373

The connectedness of friends-and-strangers graphs about graph parameters and others

Xinghui Zhao·Lihua You·Jifu Lin·Xiaoxue Zhang
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Abstract

Let $X$ and $Y$ be two graphs of order $n$. The friends-and-strangers graph $\textup{FS}(X,Y)$ of $X$ and $Y$ is a graph whose vertex set consists of all bijections $\sigma: V(X)\rightarrow V(Y)$, in which two bijections $\sigma$ and $ \sigma'$ are adjacent if and only if they agree on all but two adjacent vertices of $X$ such that the corresponding images are adjacent in $Y$. The most fundamental question about these friends-and-strangers graphs is whether they are connected. In this paper, we provide a sufficient condition regarding the maximum degree $\Delta(X)$ and vertex connectivity $\kappa(Y)$ that ensures the graph $\textup{FS}(X,Y)$ is $s$-connected. As a corollary, we improve upon a result by Bangachev and partially confirm a conjecture he proposed. Furthermore, we completely characterize the connectedness of $\textup{FS}(X,Y)$, where $X\in\textup{DL}_{n-k,k}$.

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Xinghui Zhao, Lihua You, Jifu Lin, Xiaoxue Zhang. 2025-04-01. The connectedness of friends-and-strangers graphs about graph parameters and others. https://arxiv.org/abs/2504.00373

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